In this short note we propose a new approach for the design and analysis of randomized gossip algorithms which can be used to solve the average consensus problem. We show how the Randomized Block Kaczmarz (RBK) method—a method for solving linear systems—works as gossip algorithm when applied to a special system encoding the underlying network. The famous pairwise gossip algorithm arises as a special case. Subsequently, we reveal a hidden duality of randomized gos- sip algorithms, with the dual iterative process maintaining a set of numbers attached to the edges as opposed to nodes of the network. We prove that RBK obtains a superlinear speedup in the size of the block, and demonstrate this effect through experiments.